Floss: Add metrics for Manager Client initialization

From the recent experience, bugs related to manager client
initialization are difficult to reproduce but have a high severity.

This change adds metrics to track the success rate and performance of
initialization in order to detect and address such issues earlier.

BUG=b:360983297
TEST=run device_unittests

Change-Id: I8b2e84649e16318658b79c183ba9066d65f67932
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5836464
Commit-Queue: Ying Hsu <yinghsu@chromium.org>
Reviewed-by: Ying Hsu <yinghsu@chromium.org>
Auto-Submit: Yun-hao Chung <howardchung@chromium.org>
Reviewed-by: Kyle Horimoto <khorimoto@chromium.org>
Reviewed-by: Hsin-chen Chuang <chharry@google.com>
Cr-Commit-Position: refs/heads/main@{#1353218}